Plasma Astrophysics Chapter 2: Single Particle Motion Yosuke
Mizuno Institute of Astronomy National Tsing-Hua University
Slide 2
Single Particle Motion Particle Motion in uniform B-field
Gyro-motion E x B drift Particle Motion in non-uniform B-field
Gradient drift Curvature drift Adiabatic invariants
Slide 3
Equation of Motion In dense plasma, Coulomb forces couple with
particles. So bulk motion is significant In rarefied plasma, charge
particles does not interact with other particles significantly. So
motion of each particles can be treated independently In general,
equation of motion of particle with mass m under influence of
Lorentz force is : E: electric field, B: magnetic field, q:
particles charge, v: particles velocity Which is valid for
non-relativistic motion (v
Uniform B-field: Gyration If motion is only subject to static
and uniform B field, (E=0 ) Taking a dot product with v, RHS is
zero as v B => Therefore, a static magnetic field cannot change
the kinetic energy of a particle since force is always
perpendicular to direction of motion (2.2)
Slide 5
Uniform B-field: Gyration (cont.) Decompose velocity into
parallel and perpendicular components to B: Eq (2.2) => This
equation can be split into two independent equations: These imply
that B field has no effect on the motion along it. Only affects
particle velocity perpendicular it =>
Slide 6
Uniform B-field: Cyclotron Frequency To examine the
perpendicular motion, we consider B =(0, 0, B z ) Re-write eq (2.2)
in each components: To determine the time variation of v x and v y
we take derivative eq (2.3a) & eq (2.3b), Where is the
gyrofrequency or cyclotron frequency (2.3a) (2.3b) (2.3c) (2.4a)
(2.4b)
Slide 7
Uniform B-field: Cyclotron Frequency (cont.) Gyrofrequency or
cyclotron frequency: Indicative of the field strength and the
charge and mass of particles of plasma Does not depend on kinetic
energy For electron, c is positive, electron rotates in the
right-hand sense Plasma can have several cyclotron frequencies
Figure from Schwartz et al., P.23
Slide 8
Uniform B-field: Larmor radius The v x B force is centripetal,
so Particles with faster velocities or larger mass orbit larger
radii For electron, the gyrofrequency can be written: Larmor radius
or gyroradius where B is in units of Gauss
Slide 9
Uniform B-field: Guiding Center What is path of electrons?
Solution of eqs (2.4a) & (2.4b) are harmonic: where is a
constant speed in plane of perpendicular to B Integrating, we have
Using Larmor radius (r L ), and taking a real part of above: These
describe a circular orbit at a guiding center (x 0, y 0 ) (2.5a)
(2.5b) (2.6)
Slide 10
Uniform B-field: Helical Motion In addition to this motion,
there is a velocity v z along B which is not effected by B Combine
with eq (2.6), this gives helical motion about a guiding center
Guiding center moves linearly along z with constant velocity, v ||
Pitch angle of helix is defined as guiding center motion in uniform
B-field (x 0, y 0 ) rgrg rLrL
Slide 11
Helical Motion: Magnetic Moment The charge circulates on the
plane perpendicular to B with a uniform angular frequency c Charge
passes through c /2 times per unit time This motion equivalent with
a situation that an electric current I=q c /2 is flowing in a
circular coil with radius r L It has Magnetic moment:
Therefore,
Slide 12
Uniform E & B field: E x B drift When E is finite, motion
will be sum of two motions: circular Larmor gyration + drift of the
guiding center Choose E to lie in x-z plane, so E y =0. consider
B=(0,0,B z ), Equation of motion is z-component of velocity is:
This is a straight acceleration along B. The transverse components
are:
Slide 13
Uniform E & B field: E x B drift (cont.) Differentiating
with constant E, Using we can write this as In a form similar to Eq
(2.5a) & (2.5b): Larmor motion is similar to case when E=0, but
now there is super-imposed drift v g of the guiding center in y
direction
Slide 14
Uniform E & B field: E x B drift (cont.) To obtain the
general formula for v g, we solve equation of motion As mdv/dt
gives a circular motion, already understand this effect, so set to
zero Taking cross product with B, The transverse components of this
equation are Where v E is the E x B drift velocity of the guiding
center
Slide 15
Uniform E & B field: E x B drift (cont.) vEvE
Slide 16
External force drift E x B drift of guiding center is : which
can be extended to a form for a general force F: Example: In a
gravitational field, Similar to the drift v E, in that drift is
perpendicular to both forces, but in this case particles with
opposite charge drift in opposite direction. External-force
drift
Slide 17
Drift in non-uniform field Uniform fields provide poor
descriptions for many phenomena, such as planetary fields, coronal
loops in the Sun, Tokamaks, which have spatially and temporally
varying fields. Particle drifts in inhomogeneous fields are
classified several way In this lecture, consider two drifts
associated with spatially non- uniform B: gradient drift and
curvature drift. (There are many other) In general, introducing
inhomogeneity is too complicated to obtain exact solutions for
guiding center drifts Therefore use orbit theory approximation:
Within one Larmor orbit, B is approximately uniform, i.e., typical
length-scale L over which B varies is L >> r L =>
gyro-orbit is nearly circle
Slide 18
Grad-B Drift Assumes lines of forces are straight, but their
strength is increases in y-direction Gradient in |B| causes the
Larmor radius (r A =mv/qB) to be larger at the bottom of the orbit
than at the top, which leads to a drift Drift should be
perpendicular to B and B Ions and electrons drift in opposite
directions
Slide 19
Grad-B Drift (cont.) Consider spatially-varying magnetic field,
B=(0,0,B z (y)), i,.e., B only has z-component and the strength of
it varies with y. Assume that E=0, so equation of motion is F=q(v x
B) Separating into components, The gradient of B z is => This
means that the magnetic field strength can be expanded in a Taylor
expansion for distances y < r L, (2.7a ) (2.7b ) (2.7c )
Slide 20
Grad-B Drift (cont.) Expanding B z to first order in Eqs (2.7a)
& (2.7b ) : Particles in B-field traveling around a guiding
center (0,0) with helical trajectory: The velocities can be written
in a similar form: Substituting these into eq(2.8) gives: (2.8a )
(2.8b )
Slide 21
Grad-B Drift (cont.) Since we are only interested in the
guiding center motion, we average force over a gyro-period.
Therefore in x-direction: But and => In the y-direction: Where
but (2.9a ) (2.9b )
Slide 22
Grad-B Drift (cont.) In general, drift of guiding center is
Using Eq(2.9b): So positive charged particles drift in x direction
and negative charged particles drift +x direction In 3D, the result
can be generalized to: The \pm stands for sign of charge. The
grad-B drift is in opposite direction for electrons and ions and
causes a current transverse to B Grad-B drift
Slide 23
Grad-B Drift (cont.) Below are shown particle drifts due to a
magnetic field gradient, where B(y) = zB z (y) Consider r L =mv/qB,
local gyroradius is large where B is small and is small where B is
large which gives to a drift
Slide 24
Curvature Drift When charged particles move along curved
magnetic field lines, experience centrifugal force perpendicular to
magnetic field lines Assume radius of curvature (R c ) is >>
r L The outward centrifugal force is This can be directly inserted
into general form for guiding-center drift => Drift is into or
out of page depending on sign of q Curvature drift
Slide 25
Adiabatic invariance of magnetic moment Gyrating particle
constitutes an electric current loop with a dipole moment: The
dipole moment is conserved, i.e., is invariant. Called the first
adiabatic invariant. = const even if B varies spatially or
temporally. If B varies, then v perp varies to keep = const =>v
|| also changes. It gives rise to magnetic mirrorng. Seen in
planetary magnetospheres, coronal loops etc.
Slide 26
Magnetic mirroring Consider B-field is in z-direction and whose
magnitude varies in z- direction. If B-field is axisymmetric, B =0
and d/d =0 (r, , z) This has cylindrical symmetry, so write How
does this configuration gives a force that can trap a charged
particle? Can obtain B r from. In cylindrical coordinates: If is
given at r=0 and does not change much with r, then (2.10 )
Slide 27
Magnetic mirroring (cont.) Now have B r in terms of B z, which
use to find Lorentz force on particle. The components of Lorentz
force are: As B =0, two terms vanish and terms (1) & (2) give
Larmor gyration. Term (3) vanishes on the axis and cause a drift in
radial direction. Term (4) is therefore the one of interest
Substitute from eq (2. 10): (1) (2) (3) (4)
Slide 28
Magnetic mirroring (cont.) Averaging over one gyro-orbit, and
putting & r = r L This is called the mirror force, where -/+
arises because particles of opposite charge orbit the field in
opposite directions. Above is normally written: or where In 3D,
this can be generalized to: where F || is the mirror force parallel
to B and ds is a line element along B
Slide 29
Adiabatic invariants Symmetry principles: Periodic motion
conserved quantity symmetry conserved law What if the motion is
almost periodic? Hamiltonian formulation: q and p is canonical
variables and motion almost periodical => Adiabatic invariant is
a property of physical system that stays constant when changes
occur slowly is constant, called adiabatic invariant
Slide 30
First adiabatic invariant As particle moves into regions of
stronger or weaker B, Larmor radius changes, but remains invariant
To prove this, consider component of equation of motion along B:
Multiplying by v || : Then, The particle energy must be conserved:
Using (2.11 )
Slide 31
First adiabatic invariant (cont.) Use eq(2.11), As B is not
equal to 0, this implies that: That is = const in time (invariant)
is known as the first adiabatic invariant of the particle orbit. As
a particle moves from a weak-field to a strong-field region, it
sees B increasing and therefore v perp must increase in order to
keep constant. Since total energy must remain constant, v || must
decrease. If B is high enough, v || essentially => 0 and
particle is reflected back to weak-field
Slide 32
Consequence of invariant Consider B 0 & B 1 in the weak-
and strong-field regions. Associated speeds are v 0 & v 1 The
conservation of implies that So as B increases, the perpendicular
component of particle velocity increases => particles move more
and more perpendicular to B However, since E=0, the total particle
energy cannot increase. Thus as v perp increases, v || must
decrease. The particle slow down in its motion along the filed If
field is strong enough, at some point the particle may have v ||
=0
Slide 33
Consequence of invariant cont. At B 1, v 1,|| =0, From
conservation of energy: Using we can write But, where is the pitch
angle Therefore Particle with smaller will mirror in regions of
higher B. If is too small, B 1 >> B 0 and particle does not
mirror B v || v v perp
Slide 34
Consequence of invariant cont.2 Mirror ratio is defined as The
smallest of a confined particle is This defines region in velocity
space in the shape of a cone, called the loss cone. If a particle
is in a region between two high field. The particle may be
reflected at one, travel towards the second, and also reflect
there. Thus the particle motion is confined to a certain region of
space, this process is known as magnetic trapping.
Slide 35
Other adiabatic invariants Second adiabatic invariant:
longitudinal invariant of a trapped particle in the magnetic mirror
This property is used in Fermi acceleration. Third adiabatic
invariant: flux invariant, which means the magnetic flux through
the guiding center orbit is conserved
Slide 36
Application for Astrophysics Particle trapping: due to magnetic
mirroring in the Earths van Allen radiation belt and energetic
electrons confined in solar coronal magnetic loops Particle
transport of energetic particles (galactic cosmic-rays, solar
energetic particles, ultra-relativistic particles from
extragalactic sources): governed by variety of particle drifts
Particle acceleration: due to electric fields in pulsar
magnetosphere and solar flares. Magnetic mirroring due to either
stochastic motion of high field regions, or systematic motion
between converging mirrors in the vicinity of shock waves, leads to
Fermi acceleration of particles Radiation: given by relativistic
gyrating particles (synchrotron radiation) or accelerating
particles (bremsstrahlung)
Slide 37
Summary of single particle motion Charged particle motion in B
& E fields is very unique and has interesting physical
properties Particle Motion in uniform B-field (no E-field)
Gyration: gyro (cyclotron)-frequency, Larmor radius, guiding center
Particle Motion in uniform E & B-fields E x B drift Particle
Motion in non-uniform B-field Grad-B drift, Curvature drift
magnetic mirroring Adiabatic invariants Magnetic moment